Interfacial instabilities in two-dimensional Stokes flow: a weakly nonlinear analysis

Abstract

Two-dimensional Stokes flow with injection and suction is investigated through a second-order, perturbative mode-coupling approach. We examine the time-dependent disturbance of an initially circular interface separating two viscous fluids, and derive a system of nonlinear differential equations describing the evolution of the interfacial perturbation amplitudes. Linear stability analysis reveals that an injection-induced expanding interface is stable, while a contracting motion driven by suction is unstable. Curiously, at the linear level this suction instability is independent of the viscosity contrast between the fluids. However, second-order results tell a different story, and show that the viscosity contrast plays a key role in determining the shape of the emerging interfacial patterns. By focusing on the onset of nonlinear effects, we have been able to extract valuable information about the most fundamental features of the pattern-forming structures for arbitrary values of viscosity contrast and surface tension.